Question

question solve for x using the master product. $2x^2 - 11x - 6 = 0$ master product:

Explanation:

Step1: Identify \(a\), \(b\), \(c\)

For \(2x^2 - 11x - 6 = 0\), \(a = 2\), \(b=-11\), \(c = -6\).
Find two numbers \(m\) and \(n\) such that \(m\times n=a\times c = 2\times(-6)=-12\) and \(m + n=b=-11\). The numbers are \(-12\) and \(1\) (since \(-12\times1=-12\) and \(-12 + 1=-11\)).

Step2: Rewrite the middle term

Rewrite \(-11x\) as \(-12x+x\):
\(2x^2-12x + x-6 = 0\)

Step3: Factor by grouping

Group the first two and last two terms:
\((2x^2-12x)+(x - 6)=0\)
Factor out common factors from each group:
\(2x(x - 6)+1(x - 6)=0\)
Factor out \((x - 6)\):
\((2x + 1)(x - 6)=0\)

Step4: Solve for \(x\)

Set each factor equal to zero:
\(2x+1 = 0\) or \(x - 6=0\)
For \(2x+1 = 0\), \(2x=-1\), \(x=-\frac{1}{2}\)
For \(x - 6=0\), \(x = 6\)

Answer:

\(x = 6\) or \(x=-\frac{1}{2}\)